as a diagonal Coulomb blockade peak-ridge (red
arrows). The slope of this ridge determines how
to compensate the wire gates (Vg2,g3) with the
junction gate (Vg1). Data can then be taken in the
cotunneling regime for an effectively constant
mdot. Tunneling spectra measured along the red
line in Fig. 3B at various fields are shown in Fig.
3, C to F. A pair of ABSs that moves with mwire
can be seen at B = 0 (Fig. 3C). The spectrum is
symmetric around zero Vsd, reflecting particle-hole symmetry. The minimum energy of the
ABS is z = 130 meV, which is smaller than the
effective gap D* = 220 meV. The pair of ABSs
splits into two pairs when the applied magnetic field lifts the spin degeneracy, visible
above B = 0.4 T (Fig. 3D). The low field splitting corresponds to an effective g-factor, g* 4
(the g*-factor estimated from the ABS-energy/
magnetic field slope may differ considerably
from the intrinsic g*-factor). At higher magnetic fields, the inward ABSs cross at zero and
reopen, forming a characteristic oscillatory pattern (Fig. 3, E and F). The gap reopening at
more positive Vg2,g3 is relatively slow, leading
to a single zero-bias peak in the range of Vg2,g3

5.8 to 7 V (Fig. 3F).

The magnetic field dependence of the ABSspectrum near the ABS energy minimum isshown in Fig. 3G. The evolution of the ABSscan be clearly followed: They split at low field,the inner ABSs merge around B = 1 T, theysplit again at higher fields, and the resplit ABSsmerge with the higher-energy ABSs above B =

1.7 T. Here, the emergence of a zero-bias peak
and its splitting is qualitatively similar to the
observations reported in (27, 30). However, the
B-dependent ABS spectrum at more positive gate
voltage (Fig. 3H) shows a merging-splitting-merging behavior, giving rise to an eye-shaped
loop between 1 and 2 T. At even more positive
gate voltage (Fig. 3I), the spectrum displays an
unsplit zero-bias peak from 1.1 to 2 T. The first
excited ABSs in Fig. 3, G to I, are still visible at a
high magnetic field—for instance, as marked at
B = 1.2 T in Fig. 3, H and I. Qualitatively, the
lowest-energy ABSs in Fig. 3, H and I, tend to
split after crossing but are pushed back by the
first excited ABSs, resulting in either a narrow
splitting or an unsplit zero-bias peak. The measurements in Fig. 3, C to I, were taken in an
even Coulomb valley of the end dot, but the
qualitative behavior does not depend on end-dot parity. Similar results measured in an odd
valley of the end dot are provided in (35).

The different field dependences of the subgap
states—either crossing zero or sticking at zero—
can be understood as reflecting a transition from
ABS to MBS (14, 17). For the ABSs in the regime
of Vg2,g3 < 5.8 V, their crossing in Zeeman field is
a signature of parity switching, similar to ABSs
in a quantum dot, such as investigated in (30). In
contrast, behavior in the range of Vg2,g3 5.8 to

7 V indicates that the system is in the topo-logically nontrivial regime, with MBS levels thatstick to zero as the magnetic field increases. In afinite-size wire, SOI induces anticrossings betweendiscrete ABSs, thus pushing levels to zero, pre-venting further splitting. We ascribe the differ-ences in the qualitative behavior in Fig. 3, G toI, to state-dependent SOI-induced anticrossings,which depend on gate voltage. The excited ABSin Fig. 3G and the ones in Fig. 3, H and I, arepresumably not the same state, but belong todifferent subgap modes [investigated in detailin (35)].

For a long wire, the topological phase transition is marked by a complete closing and
reopening of a gap to the continuum, with a
single discrete state remaining at zero energy
after the reopening. For a finite wire, the continuum is replaced by a set of discrete ABSs,
and at the transition where a single state becomes
pinned near zero energy, there remains a finite
gap d to the first discrete excited state. At this
transition point (where the gap of the corresponding infinite system would close and
its spectrum would be linear), Ek ¼ Rajkj, where
R is a renormalization factor due to the strong
coupling between the semiconducting wire and
its superconducting shell (35), a is the spin-orbit
coupling strength, and k is the electron wave
vector. From this relation, we can connect d to
the ratio L/x as L/x ≈ RpD′/d, where L is the
separation between Majoranas (the wire length
in the clean limit), x is the effective superconducting coherence length near the topological

0 21

B ( T)

0 21
0 0.02

0 0.02

3.0 3.6
0 0.03

0 0.02

17.6 18.0
Vg1 (V)

Vs d(mV)0.3

0.0

-0.3 3.0 3.6

0 0.08 B = 0.5 T,Vbg= -2.5 V B = 1 T,Vbg= -2.5 V

Vsd(mV)0.4

0.0

-0.4

17.6 18.0

Vg1 (V)

0 0.03 B = 0.5 T,Vbg= -7 V B = 1 T,Vbg= -7 V

Vg1 = 3.25 V, Vbg= -2.5 V

Vg1 = 17.85 V, Vbg= -7 V

dI/d
V (e
2/ h
)

dI/dV
(e
2/ h
)

-0.4 -0.2 0.0 0.2 0.4

-0.3 0.3 0.0
0.00

0.00
0.04
0.04

0 T
0.5 T
0.5 T

1 T
1.5 T
2 T
0 T
1 T
1.5 T
2 T

Vsd (mV)

Vsd (mV)

dI/dV (e2/h) dI/dV (e2/h) dI/dV (e2/h)

dI/dV (e2/h) dI/dV (e2/h) dI/dV (e2/h)

Fig. 2. Tunneling spectra for large and zero ABS density. (A) Differential
conductance measured for device 1 as a function of Vsd and Vg1, measured at

B = 0.5 Tand Vbg = –2.5 V, Vg2,g3 = –10 V. The white arrows indicate Zeemansplit dot levels. (B) The same as (A), but at B = 1 T. ABSs can be clearlyidentified below the superconducting gap. (C) Differential conductance as afunction of Vsd and B (B-Vsd sweep), measured at the gate voltage indicatedby the white lines in (A) and (B). The triangle and square indicate at whichfields (A) and (B) are measured, respectively. Blurring of data in narrow B < 0range is due to heating caused by sweeping field away from zero. (D) Line-cuttaken from (C) at various B values. Lines are offset by 0.01 e2/h each forclarity. The conductance peaks below the superconducting gap indicate thatthe wire is in a subgap-state–rich regime. A well-defined zero-bias peak canbe seen at high field. (E to H) Similar to (A) to (D), but measured at Vbg = –7 Vand Vg2,g3 = –10 V, and with (G) measured at the gate voltage indicated by theblack lines in (E) and (F). The diamond and circle indicate at which fields (E)and (F) are measured, respectively. Here, a hard superconducting gap is clearlyseen, with a critical magnetic field Bc up to ~2.2 T. No subgap structure isobserved across the full range of field, 0 to 2 T.